The Sierpi\'{n}ski Domination Number

TL;DR

This paper introduces the Sierpiński domination number, analyzing its bounds and exact values for the Sierpiński product of two cycles, expanding understanding of domination parameters in complex graph constructions.

Contribution

The paper defines the Sierpiński domination number, establishes bounds, and computes exact values for the product of two cycles, providing new insights into domination in Sierpiński graphs.

Findings

Established bounds for the Sierpiński domination number.

Determined the exact upper Sierpiński domination number for two cycles.

Partially determined the lower Sierpiński domination number for two cycles.

Abstract

Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpi\'{n}ski product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. In this paper, we define the Sierpi\'{n}ski domination number as the minimum of $\gamma(G\otimes _f H)$ over all functions $f \colon V(G)\rightarrow V(H)$. The upper Sierpi\'{n}ski domination number is defined analogously as the corresponding maximum. After establishing general upper and lower bounds, we determine the upper Sierpi\'{n}ski domination number of the Sierpi\'{n}ski product of two cycles, and determine the lower Sierpi\'{n}ski domination number of the Sierpi\'{n}ski product of two cycles in half of the cases and in the other half cases restrict it to two values.